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Numerical Conformal Mapping to One-Tooth Gear-Shaped Domains and Applications

Philip R. Brown, R. Michael Porter

Abstract

We study conformal mappings from the unit disk (or a rectangle) to one-tooth gear-shaped planar domains from the point of view of the Schwarzian derivative, with emphasis on numerical considerations. Applications are given to evaluation of a singular integral, mapping to the complement of an annular rectangle, and symmetric multitooth domains.

Numerical Conformal Mapping to One-Tooth Gear-Shaped Domains and Applications

Abstract

We study conformal mappings from the unit disk (or a rectangle) to one-tooth gear-shaped planar domains from the point of view of the Schwarzian derivative, with emphasis on numerical considerations. Applications are given to evaluation of a singular integral, mapping to the complement of an annular rectangle, and symmetric multitooth domains.

Paper Structure

This paper contains 23 sections, 10 theorems, 70 equations, 10 figures.

Table of Contents

  1. Introduction
  2. One-tooth gear domains
  3. Conformal mapping from disk to a one-tooth gear
  4. Conformal mapping from rectangle to gear
  5. Relation between disk and rectangle Schwarzians
  6. Formulas for calculating a gear mapping
  7. Numerical computation of gear parameters
  8. Auxiliary second-order linear ODE
  9. Computational procedures for the disk
  10. Calculation via the 2-jet.
  11. Calculation via the spectral parameter power series.
  12. Computational procedure for the rectangle
  13. Condition to make pregear into a gear
  14. Computational procedure
  15. Numerical results
  16. ...and 8 more sections

Key Result

Proposition 2.1

(a) Let $D$ be a circular quadrilateral with the above symmetries, having two interior angles equal to $\pi/2$ and two interior angles equal to $3\pi/2$. Assume that one tooth edge of $D$ lies in the upper and the other in the lower half-plane. Then $D$ is a pregear if and only if the full circles $

Figures (10)

  • Figure 1: $|\varphi_{\tau,\mu}|$ for Schwarzian derivative (\ref{['eq:varphi']})
  • Figure 2: Structure of pregears as in the proof of Proposition \ref{['prop:pregeartogear']}. Note that precisely one of $b^{\pm}$ is interior to the pregear.
  • Figure 3: Pairs $(\gamma,\beta)$ for fixed $t=\pi/n$ with $n=3,4,\dots,10$. Larger dashing indicates larger value of $t$.
  • Figure 4: Graphs of the curvature $\kappa(\lambda)$ of the tooth edges of the family of pregears corresponding to $R_{t\lambda}$ for $t/\pi=0.1$, $0.2$, $0,3$, $0,4$.
  • Figure 5: Graphs of $\beta(\lambda)$ and $\gamma(\lambda)$ produced by SPPS formulas, for $t=\pi/4$.
  • ...and 5 more figures

Theorems & Definitions (11)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 4.1
  • Conjecture 5.1
  • Theorem 7.1
  • ...and 1 more